This novel gravitational theory generalizes \(\mathrm {GR}\)  by
including higher-order matter ingredients of the type
\(T_{\alpha \beta }T^{\alpha \beta }\equiv \mathbf {T}^{2}\)  in the
\(\mathrm {GR}\) 's generic action, contrary to the theories that
include higher-order curvature ingredients, such as \(f(\mathbf {R})\) 
and \(f(\mathcal {G})\)  models of gravity. When we generalize
\(\mathrm {GR}\)  by accepting higher-order matter stresses, i.e., the
scalar square of the stress-energy tensor to the action, we may find
that the cosmological behavior has a deeper structure. We have
investigated in detail the cosmological effects by including matter
stresses of higher-order under the \(\mathbb {Q}_{\mathbf {H}}\) 
evolution plus the vanishing \(\mathbb {C}_{\mathbf {F}}\)  condition. In
order to generate a few particular forms of solutions, we have
considered further restrictions on the structural variables in Sec.
VI. One of these restrictions has been proved to be
especially useful for characterizing the development of a matter
configuration with a void encircling the center. All the theoretical
frameworks governing the spherical star have been formulated and
various analytical solutions under the above-stated conditions have
been provided. Few of the presented cosmological solutions follow
the Darmois constraints on and thereby exhibiting shells either
boundary-surfaces. Other cosmological models are procured when
Darmois constraints are relaxed and Israel constraints are applied
across the shells. We have examined both dissipating and
non-dissipating spherical gravitational structures. In the
dissipative case, by relaxing the homologous condition (studied in
[1]}) and adopting a less restrictive
\(\mathbb {Q}_{\mathbf {H}}\)  condition defined in Sec. IV, we
have obtained a large number of analytical solutions satisfying the
zero \(\mathbb {C}_{\mathbf {F}}\)  constraint, contrary to the unique
\(\mathrm {FRW}\)  model corresponding to the homologous constraint.
These temperature expressions comprehend the thermal history of the
gravitational objects.
